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Baltic Way
2016 Baltic Way
9
System of equations
System of equations
Source: Baltic Way 2016, Problem 9
November 5, 2016
algebra
system of equations
Problem Statement
Find all quadruples
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
of real numbers that simultaneously satisfy the following equations:
{
a
3
+
c
3
=
2
a
2
b
+
c
2
d
=
0
b
3
+
d
3
=
1
a
b
2
+
c
d
2
=
−
6.
\begin{cases} a^3 + c^3 = 2 \\ a^2b + c^2d = 0 \\ b^3 + d^3 = 1 \\ ab^2 + cd^2 = -6.\end{cases}
⎩
⎨
⎧
a
3
+
c
3
=
2
a
2
b
+
c
2
d
=
0
b
3
+
d
3
=
1
a
b
2
+
c
d
2
=
−
6.
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